Gamma Function of One Half/Proof 3

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Theorem

$\map \Gamma {\dfrac 1 2} = \sqrt \pi$


Proof

\(\ds \map \Gamma {\dfrac 1 2}\) \(=\) \(\ds \int_0^{\to \infty} t^{-\frac 1 2} e^{-t} \rd t\) Definition of Gamma Function
\(\ds \) \(=\) \(\ds \int_0^{\to \infty} u^{-1} e^{-u^2} 2 u \rd u\) Integration by Substitution, $\map \phi u = u^2$
\(\ds \) \(=\) \(\ds 2 \int_0^{\to \infty} e^{-u^2} \rd u\) Linear Combination of Definite Integrals
\(\ds \) \(=\) \(\ds \int_{\to -\infty}^{\to \infty} e^{-u^2} \rd u\) Definite Integral of Even Function
\(\ds \) \(=\) \(\ds \sqrt \pi\) Gaussian Integral

$\blacksquare$


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